Download 6.5 Prove Triangles Similar by SSS and SAS

6.5
Prove Triangles Similar by
SSS and SAS
p Use the SSS and SAS Similarity Theorems.
Goal
Your Notes
THEOREM 6.2: SIDE-SIDE-SIDE (SSS) SIMILARITY
THEOREM
A
If the corresponding side lengths
of two triangles are
then the triangles are similar.
,
B
C
R
BC
CA
AB
, then nABC , nRST.
If } 5 } 5 }
TR
RS
ST
S
T
Use the SSS Similarity Theorem
Example 1
Is either nDEF or nGHJ similar to nABC?
H
B
8
A
When using the
SSS Similarity
Theorem, compare
the shortest sides,
the longest sides,
and then the
remaining sides.
D 8 F
4
6
E
9
12
C
16
18
24
J
G
Solution
Compare nABC and nDEF by finding ratios of
corresponding side lengths.
Shortest sides
AB
DE
}5
Longest sides
CA
FD
5
}5
5
Remaining sides
BC
EF
}5
5
, so nABC and nDEF are
The ratios are
.
Compare nABC and nGHJ by finding ratios of
corresponding side lengths.
Shortest sides
AB
GH
}5
5
All the ratios are
162 Lesson 6.5 • Geometry Notetaking Guide
Longest sides
CA
JG
}5
5
, so n
Remaining sides
BC
HJ
}5
,n
5
.
Copyright © Holt McDougal. All rights reserved.
6.5
Prove Triangles Similar by
SSS and SAS
p Use the SSS and SAS Similarity Theorems.
Goal
Your Notes
THEOREM 6.2: SIDE-SIDE-SIDE (SSS) SIMILARITY
THEOREM
A
If the corresponding side lengths
of two triangles are proportional ,
then the triangles are similar.
B
C
R
BC
CA
AB
, then nABC , nRST.
If } 5 } 5 }
TR
RS
ST
Example 1
S
T
Use the SSS Similarity Theorem
Is either nDEF or nGHJ similar to nABC?
H
B
8
A
When using the
SSS Similarity
Theorem, compare
the shortest sides,
the longest sides,
and then the
remaining sides.
D 8 F
4
6
E
9
12
16
18
C
J
24
G
Solution
Compare nABC and nDEF by finding ratios of
corresponding side lengths.
Shortest sides
8
4
AB
DE
Longest sides
CA
FD
}5 } 5 2
12
8
Remaining sides
3
2
}5 } 5 }
BC
EF
9
6
3
2
}5 } 5 }
The ratios are not all equal , so nABC and nDEF are
not similar .
Compare nABC and nGHJ by finding ratios of
corresponding side lengths.
Shortest sides
AB
GH
8
16
1
2
}5 } 5 }
Longest sides
CA
JG
12
24
Remaining sides
1
2
}5 } 5 }
BC
HJ
9
18
1
2
}5 } 5 }
All the ratios are equal , so n ABC , n GHJ .
162 Lesson 6.5 • Geometry Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Example 2
Use the SSS Similarity Theorem
Find the value of x that
makes nABC , nDEF.
E
A
20
B
x23
10
4
10
C
2x 1 3
D
F
Solution
Step 1 Find the value of x that makes corresponding side
lengths proportional.
4
4p
5 x23
Write proportion.
5
(x 2 3)
Cross Products Property
5
x2
Simplify.
5x
Solve for x.
Step 2 Check that the side lengths are proportional when
x5
.
AB 5 x 2 3 5
BC
AB
}0}
EF
DE
When x 5
DF 5 2x 1 3 5
5
✓
BC
AC
}0}
EF
5
DF
✓
, the triangles are similar by the
.
Checkpoint Complete the following exercises.
1. Which of the three triangles
are similar?
B
42
38
A
54
C
Q
2. Suppose AB is not given in nABC.
What value of AB would make
nABC similar to nQRP?
45
30
35
P
R
Y
18
X
Copyright © Holt McDougal. All rights reserved.
21
27
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Lesson 6.5 • Geometry Notetaking Guide
163
Your Notes
Use the SSS Similarity Theorem
Example 2
Find the value of x that
makes nABC , nDEF.
E
A
20
B
x23
10
4
10
C
2x 1 3
D
F
Solution
Step 1 Find the value of x that makes corresponding side
lengths proportional.
4
10
5 x23
20
Write proportion.
4 p 20 5 10 (x 2 3)
Cross Products Property
80 5 10 x 2 30
Simplify.
11 5 x
Solve for x.
Step 2 Check that the side lengths are proportional when
x 5 11 .
AB 5 x 2 3 5 8
BC
AB
}0}
EF
DE
4
10
DF 5 2x 1 3 5 25
8
20
} 5 } ✓
EF
10
25
4
10
BC
AC
}0}
} 5 } ✓
DF
When x 5 11 , the triangles are similar by the
SSS Similarity Theorem .
Checkpoint Complete the following exercises.
1. Which of the three triangles
are similar?
B
42
38
nPQR , nZXY
A
54
C
Q
2. Suppose AB is not given in nABC.
What value of AB would make
nABC similar to nQRP?
45
30
35
P
AB 5 36
Y
18
X
Copyright © Holt McDougal. All rights reserved.
R
21
27
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Lesson 6.5 • Geometry Notetaking Guide
163
Your Notes
THEOREM 6.3: SIDE-ANGLE-SIDE (SAS) SIMILARITY
THEOREM
If an angle of one triangle is
congruent to an angle of a
second triangle and the lengths
of the sides including these
,
angles are
then the triangles are similar.
M
X
P
Z
N
Y
ZX
XY
}, then nXYZ , nMNP.
5
If ∠X > ∠M, and }
PM
MN
Example 3
Use the SAS Similarity Theorem
A
Birdfeeder You are drawing a design
for a birdfeeder. Can you construct the 32 cm
top so it is similar to the bottom using
B
the angle measure and lengths shown?
C
32 cm
878
Solution
F
D
and m∠
equal 878, so
Both m∠
20 cm
20 cm
E
878
∠
>∠
. Next, compare the ratios
of the lengths of the sides that include ∠B and ∠E.
AB
5
EF
5
5
The lengths of the sides that include ∠B and ∠E
.
are
, nABC , nDEF.
So, by the
Yes, you can make the top similar to the bottom.
Checkpoint Complete the following exercise.
3. In Example 3, suppose you use equilateral triangles
on the top and bottom. Are the top and bottom
similar? Explain.
164
Lesson 6.5 • Geometry Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
THEOREM 6.3: SIDE-ANGLE-SIDE (SAS) SIMILARITY
THEOREM
If an angle of one triangle is
congruent to an angle of a
second triangle and the lengths
of the sides including these
angles are proportional ,
then the triangles are similar.
M
X
P
Z
N
Y
ZX
XY
}, then nXYZ , nMNP.
5
If ∠X > ∠M, and }
PM
MN
Example 3
Use the SAS Similarity Theorem
A
Birdfeeder You are drawing a design
for a birdfeeder. Can you construct the 32 cm
top so it is similar to the bottom using
B
the angle measure and lengths shown?
C
32 cm
878
Solution
F
D
Both m∠ B and m∠ E equal 878, so
20 cm
20 cm
E
878
∠ B > ∠ E . Next, compare the ratios
of the lengths of the sides that include ∠B and ∠E.
AB
DE
5
BC
32
8
5 } 5 }
5
20
EF
The lengths of the sides that include ∠B and ∠E
are proportional .
So, by the SAS Similarity Theorem , nABC , nDEF.
Yes, you can make the top similar to the bottom.
Checkpoint Complete the following exercise.
3. In Example 3, suppose you use equilateral triangles
on the top and bottom. Are the top and bottom
similar? Explain.
Yes, the top and bottom are similar. If the side
length of the top is a and the side length of the
bottom is b, the ratios of the side lengths are
a
} and the angles are all 608. The triangles are
b
similar by SAS or SSS.
164
Lesson 6.5 • Geometry Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
TRIANGLE SIMILARITY POSTULATE AND THEOREMS
AA Similarity Postulate If ∠A > ∠D and
∠B > ∠E, then nABC , nDEF.
AB
BC
AC
5}
5}
,
SSS Similarity Theorem If }
DE
EF
DF
then nABC , nDEF.
SAS Similarity Theorem If ∠A > ∠D and
AC
AB
} 5 } , then nABC , nDEF.
DE
DF
Example 4
To identify
corresponding
parts, redraw the
triangles so that
the corresponding
parts have the
same orientation.
Choose a method
Tell what method you would use to
show that the triangles are similar.
P
18
T
P
Q
S
12
R
Solution
Find the ratios of the lengths of the
corresponding sides.
9
24
Shorter sides
24
T
18
R
Q
9
S
12
R
Longer sides
The corresponding side lengths are
.
The included angles ∠PRQ and ∠TRS are
because they are
angles. So, nPQR , nTSR
by the
.
Checkpoint Complete the following exercise.
4. Explain how to show nJKL , nLKM.
L
56
28
Homework
16
J
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32
49
K
Lesson 6.5 • Geometry Notetaking Guide
M
165
Your Notes
TRIANGLE SIMILARITY POSTULATE AND THEOREMS
AA Similarity Postulate If ∠A > ∠D and
∠B > ∠E, then nABC , nDEF.
AB
BC
AC
5}
5}
,
SSS Similarity Theorem If }
DE
EF
DF
then nABC , nDEF.
SAS Similarity Theorem If ∠A > ∠D and
AC
AB
} 5 } , then nABC , nDEF.
DE
DF
Example 4
To identify
corresponding
parts, redraw the
triangles so that
the corresponding
parts have the
same orientation.
Tell what method you would use to
show that the triangles are similar.
P
Shorter sides
24
18
R
Q
S
12
P
18
R
Longer sides
Q
S
12
R
Solution
Find the ratios of the lengths of the
corresponding sides.
T
9
Choose a method
9
24
QR
9
3
} 5 } 5 }
4
12
RS
18
3
PR
} 5 } 5 }
4
24
RT
T
The corresponding side lengths are proportional .
The included angles ∠PRQ and ∠TRS are congruent
because they are vertical angles. So, nPQR , nTSR
by the SAS Similarity Theorem .
Checkpoint Complete the following exercise.
4. Explain how to show nJKL , nLKM.
Homework
Show that the corresponding side
lengths are proportional, then
use the SSS Similarity Theorem
to show nJKL , nLKM.
Copyright © Holt McDougal. All rights reserved.
L
56
28
32
16
J
49
K
Lesson 6.5 • Geometry Notetaking Guide
M
165